# Q.1 Determine whether each of the following relations are reflexive, symmetric and transitive:(v) Relation R in the set A of human beings in a town at a particular time given by(b) $R = \{(x,y): x\;and\;y\;live\;in\;the\;same\;locality\}$

$R = \{(x,y): x\;and\;y\;live\;in\;the\;same\;locality\}$

$\left ( x,x \right )\in R$ as $x$ and $x$ is same human being.So, it is reflexive.

$\left ( x,y \right )\in R$ means  $x\;and\;y\;live\;in\;the\;same\;locality$.

It is same as $y\;and\;x\;live\;in\;the\;same\;locality$  i.e. $\left ( y,x \right )\in R$.

So,it is symmetric.

$\left ( x,y \right ),\left ( y,z \right )\in R$ means  $x\;and\;y\;live\;in\;the\;same\;locality$ and $y\;and\;z\;live\;in\;the\;same\;locality$.

It implies that $x\;and\;z\;live\;in\;the\;same\;locality$ i.e. $\left ( x,z \right )\in R$.

Hence, it is reflexive, symmetric and
transitive.

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